A generalization of binary search
Abstract

A family of deterministic algorithms that minimizes the worstcase number of function evaluations needed to solve the (n, m)problem;

A deterministic algorithm that comes within one step of minimizing the worstcase number of parallel steps required to solve the (n,m)problem, where a given number p of concurrent function evaluations may be performed in each parallel step. This result requires that p ≤ m;

A deterministic algorithm that minimizes the expected number of function evaluations when the function f is drawn from a probability distribution satisfying a natural symmetry property;

A randomized algorithm that minimizes the worstcase expected number of function evaluations required to solve the (n, 1)problem;

Lower and upper bounds on the worstcase expected number of function evaluations required by a randomized algorithm to solve the (n, m)problem for m > 1;
All the algorithms presented in the paper are extremely simple.
The (n, m) problem is equivalent to the following natural search problem: given a table consisting of n entries in increasing order, and given keys x_{1} < x_{2} < ... < x_{m}, determine which of the given keys lie in the table. It is easily seen that the worstcase number of table entries that must be inspected in the search problem is equal to the worstcase number of function evaluations needed to solve the (n, m) problem.
Preview
Unable to display preview. Download preview PDF.
References
 [GK]C.R. Glassey and R.M. Karp, “On the Optimality of Huffman Trees,” SIAM J. Applied Math, Vol. 31, No. 2, pp. 368–378, September, 1976.CrossRefGoogle Scholar
 [HM]R. Hassin and N. Megiddo, “An Optimal Algorithm for Finding All the Jumps of a Monotone StepFunction,” J. Algorithms, Vol. 6, No. 2, pp. 265–274, June, 1985.CrossRefGoogle Scholar
 [Yao]A.C.C. Yao, “Probabilistic Computation: Towards a Unified Measure of Complexity,” Proc. 18th IEEE Symp. on Foundations of Computer Science, pp. 222–227, 1977.Google Scholar